Problem 99 - Algebra for College Students

Question

Determine whether each pair of lines is parallel, perpendicular, or neither. $$2 y=x+6, y-2 x=4$$

Step-by-Step Solution

Answer

These lines are neither parallel nor perpendicular.

Step 1: Convert Line 1 to Slope-Intercept Form

The first line is given by the equation \(2y = x + 6\). To convert it to the slope-intercept form \(y = mx + b\), divide both sides by 2 to get:\(y = \frac{1}{2}x + 3\). Here, the slope \(m_1\) is \(\frac{1}{2}\).

Step 2: Convert Line 2 to Slope-Intercept Form

The second line is given by the equation \(y - 2x = 4\). To convert it to the slope-intercept form \(y = mx + b\), add \(2x\) to both sides to get:\(y = 2x + 4\). Here, the slope \(m_2\) is \(2\).

Step 3: Compare the Slopes

The slope of the first line \(m_1\) is \(\frac{1}{2}\), and the slope of the second line \(m_2\) is \(2\). To determine if the lines are parallel, perpendicular, or neither:1. If \(m_1 = m_2\), the lines are parallel.2. If \(m_1\cdot m_2 = -1\), the lines are perpendicular.3. If neither condition is true, the lines are neither parallel nor perpendicular.Given \(m_1 = \frac{1}{2}\) and \(m_2 = 2\):- \(\frac{1}{2} eq 2\), so they are not parallel.- \(\frac{1}{2} \times 2 = 1\), so they are not perpendicular.

Key Concepts

slope-intercept form

The slope-intercept form is a way to write linear equations. It makes identifying the slope and y-intercept easy. The general formula is written as follows: \ \[ y = mx + b \]. \ Here, \( m \) is the slope of the line. The slope tells us how steep the line is. \( b \) is the y-intercept. This is where the line crosses the y-axis. \ To convert a line equation into this form, we need to solve for \( y \). Let’s take an example from our exercise. For the equation \( 2y = x + 6 \), we divide everything by 2: \ \ \[ y = \frac{1}{2}x + 3 \]. \ Now, it's in slope-intercept form. And we can easily see the slope \( m = \frac{1}{2} \) and the y-intercept \( b = 3 \). \

slope comparison

When comparing lines, the slope is key. The slope shows the direction and steepness of a line. By comparing slopes, we can tell if lines are parallel or perpendicular. \ Here are the steps:

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• Parallel lines have the same slope. If \( m_1 = m_2 \), the lines are parallel.
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• Perpendicular lines have slopes that are negative reciprocals. If the product of the slopes \( m_1 \cdot m_2 = -1 \), the lines are perpendicular.
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\ For example, we found that the slopes in our exercise were \( \frac{1}{2} \) and \( 2 \). Let’s compare them:

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• If \( \frac{1}{2} = 2 \), they’re parallel. But they aren’t.
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• Next, check if their product is \( -1 \). Here, \( \frac{1}{2} \cdot 2 = 1 \), not \( -1 \). So, they’re not perpendicular either.

\ By comparing slopes, we concluded these lines are neither parallel nor perpendicular.

line equations

Understanding line equations is vital in geometry. They help us describe straight lines using mathematical language. A line equation can take different forms, but the most common is the slope-intercept form: \ \[ y = mx + b \]. \ Another common form is the standard form \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants. Converting between different forms can be handy. In our exercise:

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• We started with \( 2y = x + 6 \) and converted it into \( y = \frac{1}{2} x + 3 \).
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• The other equation \( y - 2x = 4 \) was converted to \( y = 2x + 4 \).

Recognizing these forms helps identify slopes quickly. It also simplifies solving problems related to parallelism and perpendicularity.

So next time you see a line equation, try converting it to slope-intercept form. It will make things a lot clearer!